I have a model following the PDF $$f_\theta(x) = \theta^2 x \exp(-\theta x)\delta_{[0, +\infty)}(x)$$ where $\theta > 0$ is the parameter and $\delta_\mathcal{S}$ denotes the indicator function of the set $\mathcal{S}$.
I am ask to prove that this model is regular, for which I have to prove that it is dominated. How should I do that?
Below is the definition of "dominated" that I found the the book Theory of Point Estimation by E.L. Lehmann and George Casella:
Statistical problems are concerned not with single probability distributions but with families of such distributions $$\mathcal{P} = \{P_\theta, \theta\in\Omega\}$$ defined over a common measurable space $(\mathcal{X}, \mathcal{A})$. When all the distributions of $\mathcal{P}$ are absolutely continuous with respect to a common measure $\mu$, as will usually be the case, the family $\mathcal{P}$ is said to be dominated (by $\mu$).
Most of the examples with which we shall deal belong to one or the other of the following two cases.
(i) The discrete case. Here, $\mathcal{X}$ is a countable set, $\mathcal{A}$ is the class of subsets of $\mathcal{X}$, and the distributions of $\mathcal{P}$ are dominated by counting measure.
(ii) The absolutely continuous case. Here, $\mathcal{X}$ is a Borel subset of a Euclidean space, $\mathcal{A}$ is the class of Borel subsets of $\mathcal{X}$, and the distributions of $\mathcal{P}$ are dominated by Lebesgue measure over $(\mathcal{X}, \mathcal{A})$.
Thank you in advance for any help!
